Functional Steiner symmetrization has turned out to be very fruitful in proving isoperimetric theorems in analysis and function theory (see. e.g., [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, and the references therein). In the beautiful paper [7]}, Cianchi and Fucso analyzed the case of equality in Steiner symmetrization inequalities for Dirichlet-type integrals. Specially, Cianchi and Fucso [7]} proved the following Pólya-Szegö principle for \(f\in W_0^{1,1}(G)\) :
\(\int _{SG}\psi (\nabla Sf)dx\le \int _{G}\psi (\nabla f)dx,\) 
